After six years of open bitterness and hostility between their two nations, President Reagan and Soviet leader Mikhail Gorbachev appeared together on a stage here today to express a joint determination to curb the arms race and to "improve U.S.-Soviet relations and the international situation as a whole."

But the two leaders, concluding three days of summitry marked by outward appearances of warm personal contact between them, were unable to elaborate any new specific steps they will take to limit or reduce their nuclear arsenals. Instead, they celebrated the reaching of accords on cultural and other bilateral matters, and emphasized their new agreement on the need for a fresh start in East-West relations.

Standing before huge American and Soviet flags draped at the back of the stage of the International Conference Center that has been the headquarters for nearly 3,000 journalists covering the summit, Reagan said he was ready for the "hard work ahead" in improving Soviet-American relations.

"General Secretary Gorbachev," the president said, "we ask you to join us in getting the job done, as I'm sure you will."

Gorbachev, speaking at a separate, televised press conference he held shortly after bidding farewell to Reagan, voiced the judgment that "the world has become a more secure place" as a result of the meeting.

But the Soviet leader also frankly acknowledged in his press conference that sharp exchanges had marked some of the private meetings, and he conceded that he had not been able to diminish in any way Reagan's commitment to the Strategic Defense Initiative (SDI), the space weapons system that Gorbachev said would cause "all restraints [to] be blown to the winds" in nuclear competition.

"The American side turned out not to be ready for any big basic arms control decisions" at the summit, Gorbachev asserted in remarks that contrasted to his uniformly positive presentation at the joint ceremony.

There was no public sign that Gorbachev had won a pledge from Reagan to extend observance of the unratified SALT II strategic arms treaty, which had been an important Soviet goal, according to Soviet officials.

Reagan did not appear before journalists after the ceremony, preferring to take his message directly to a joint session of Congress that he addressed shortly after arriving in Washington tonight. There, he differed with Gorbachev's pessimistic assessment of progress on arms control and on which side bore the responsibility for the lack of more progress in that area.

"I can't claim we had a meeting of the minds on such fundamentals as ideology or national purpose, but we understand each other better. That's the key to peace," Reagan told Congress.

"We moved arms control forward from where we were," the president added, saying that the United States had succeeded in "turning the talks toward our chief goal, offensive reductions."

A senior U.S. official accompanying Reagan on the return to the United States said that no common ground on SDI had been established at the summit and added that neither side offered a definition of what it meant by "research," which Reagan has said he will not abandon.

Both leaders left this Swiss lakeside city this afternoon to report on the summit to their allies before returning home. Reagan flew to Brussels, where he landed in a light snow and spent 20 minutes briefing a generally enthusiastic audience of leaders of North Atlantic Treaty Organization nations.

Italian Prime Minister Bettino Craxi said Reagan had noted that he found Gorbachev "more flexible and more reasonable" than he had expected. The NATO secretary general, Lord Carrington declared that "Geneva is not the end of a process but, we hope, the beginning of a new and more constructive stage."

Gorbachev flew to Prague, where he met with the leaders of the Soviet Union's Eastern European allies in the Warsaw Pact, who joined in describing the summit as "useful" and "highly important" after hearing Gorbachev's report.

The 31-paragraph joint statement that was distributed at the morning closing ceremony contained phrases that each side could point to as justifying the extensive, competing public diplomacy campaigns that preceded the summit.

The Soviets obtained a reaffirmation of the Jan. 8 joint agreement signed in Geneva to "prevent an arms race in space and to terminate it on Earth."

Agreement for regular summit meetings and other high-level contacts also echoed Soviet calls for a return to a relationship closer to the detente period of the 1970s. The statement confirmed that Gorbachev had agreed to visit the United States "in the nearest future," which officials said meant the summer of 1986. Reagan agreed to go to Moscow for a return visit.

Among the U.S. ideas the Soviets seemed to have accepted was a commitment in the joint statement to "accelerate the work" at the Geneva negotiations on strategic and intermediate-range ballistic missiles and space systems. The leaders also called for early progress on "the principle of 50 percent reductions" in both nuclear arsenals, and endorsed the idea of an interim intermediate-range missile agreement.

"The sides," the statement declared, "have agreed that a nuclear war cannot be won and must never be fought . . . . They will not seek to achieve military superiority."

The language of the statement represented something of a change in Reagan's long-term view of competition with the Soviet Union, which he called "the focus of evil" two years ago.

The Republican platform in 1980, when Reagan was first elected, included a commitment to U.S. military superiority. But Reagan dropped this position in 1984 and said he favored "parity" between the superpowers.

The Soviets, for their part, seemed to shift on substance. They agreed in the statement to an improvement of relations without getting any specific limitations in Reagan's nuclear rearmament program, as they once had demanded.

Final American agreement to a joint statement, a favored diplomatic tactic of the Russians but one for which American officials had not voiced much enthusiasm, was reportedly in doubt until a few hours before the closing ceremony began.

On Nov. 9, Reagan's national security adviser, Robert C. McFarlane, told reporters that a final communique, which usually summarizes points of agreement, was "unlikely and probably an inappropriate measure of the meeting." Three days later, Reagan told European reporters that he was not "a great fan of communiques."

Despite these public assertions, U.S. officials said privately before departing for Geneva that they expected a joint statement to be issued and they discussed the issue with the Soviets on the first day of the summit. Sources said that both sides agreed in principle on a joint statement yesterday but that the agreement nearly unraveled early this morning because of disagreements that centered on SDI.

U.S. sources said the Soviets had wanted a more explicit condemnation of what they have called "space strike weapons" in the joint statement and that this almost became a sticking point. The joint statement was agreed to at 4:15 a.m., these sources said, after U.S. negotiators said they would insist on separate statements if the Soviets demanded an explicit reference to the Strategic Defense Initiative.

But Gorbachev made it clear that he has not changed his mind about the dangers of the U.S. missile defense plan, even though he has accepted Reagan's request to call the proposal "SDI" rather than "Star Wars."

At his news conference, Gorbachev said the Soviets were prepared to make radical reductions in offensive nuclear weapons "only if the door to weapons in space is firmly slammed shut."

White House spokesman Larry Speakes said Gorbachev made "forceful points on SDI" in the five hours of private discussions the Soviet leader held with the president during their two-day summit.

The section in the joint statement on nuclear arms and space talks echoed the declaration signed by Secretary of State George P. Shultz and then-foreign minister Andrei Gromyko in Geneva Jan. 8, when they pledged that the United States and the Soviet Union would attempt "to prevent an arms race in space and terminate it on earth, to limit and reduce nuclear arms and enhance strategic stability."

U.S. officials accompanying Reagan back to Washington on the 8 1/2-hour flight from Brussels told reporters that they had been intrigued by the mild tone of the Soviet leader's remarks about Afghanistan. In contrast, these officials described the conversations on human rights as among the most heated of the summit.

In his speech to Congress, Reagan placed particular emphasis on the cultural agreement, which he said was "designed to bring the best of America's artists and academics to the Soviet Union" and to "expand the opportunities for Americans to experience the Soviet people's rich cultural heritage."

The other agreements contained in a document signed at the Geneva ceremony by Shultz and Foreign Minister Eduard Shevardnadze as the two leaders looked on cover air safety in the North Pacific, the opening of a U.S. consulate in Kiev and a Soviet consulate in New York, and research on solar technology. They agreed to emphasize the "potential importance of the work aimed at utilizing controlled thermonuclear fusion."

Throughout the final ceremony, Reagan and Gorbachev appeared both friendly and businesslike as they chatted with one another through interpreters and read individual statements. They warmly applauded each other, then adjourned for their sixth and last private meeting of the summit to toast with champagne their colleagues who had worked out the statement.

Georgi Arbatov, one of the most prominent Soviet experts on the United States, was asked at this morning's ceremony whether the summit had lived up to the idea of the "fresh start" that Reagan said had been achieved.

"Well, it could have been fresher," Arbatov replied. "But it was okay."